Quotients (Topology)

The quotient of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes

The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the equivalence map.

The equivalence class of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.

An equivalence relation $$∼$$ on a set $$X$$ is a subset $$S$$ of $$X × X$$ (where $$(a,b) ∈ S$$ is written $$a ∼ b$$) that satisfies the following for all $$a,b,c \in X$$:

1. Reflexive: $$a ∼ a$$
2. Symmetric: $$a ∼ b$$ implies $$b ∼ a$$
3. Transitive: $$a ∼ b$$ and $$b ∼ c$$ implies $$a ∼ c$$

FBT = Function Building Theorem for quotient sets: Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:

1. There is a well-defined function $$g:X/∼ → Y$$ defined by $$g([x]) = f(x)$$ for all in $$X/∼$$; that is, $$g ∘ q = f$$, where $$q$$ is the equivalence map.
2. If $$f$$ is onto, then $$g$$ is onto.
3. If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.

The quotient topology, or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a quotient space of $$X$$, and the equivalence map $$q$$ is called the quotient map induced by $$∼$$.